2017/02/23 by Mustafa Cengiz, Cengiz, Mustafa
Mathematics · Computer Science · #Geometric and Algebraic Topology #semigroups and automata theory #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1702.07115
An open book decomposition of a 3-manifold M induces a Heegaard splitting for M, and the minimal genus among all Heegaard splittings induced by open book decompositions is called the open book genus of M. It is conjectured by Ozbagci \citeO that the open book genus is additive under the connected sum of 3-manifolds. In this paper, we prove that a non-prime 3-manifold which has open book genus 2 is homeomorphic to L(p,1)#L(q,1) for some integers p,q≠±1, that is, it has non-trivial prime pieces of open book genus 1. In particular, there cannot be a counter-example to additivity of the open book genus such that the connected sum has open book genus 2.